Many branches of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
study objects of a given type and prove a
classification theorem
In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class.
A few issues relate ...
. A common theme is that the classification results in a number of series of objects and a finite number of exceptions — often with desirable properties — that do not fit into any series. These are known as exceptional objects. In many cases, these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics often relate to the exceptional objects in others.
A related phenomenon is
exceptional isomorphism In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ''a'i'' and ''b'j'' of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such is ...
, when two series are in general different, but agree for some small values. For example,
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
s in low dimensions are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to other
classical Lie group
In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or ske ...
s.
Regular polytopes
The prototypical examples of exceptional objects arise in the classification of
regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
s: in two dimensions, there is a series of
regular ''n''-gons for ''n'' ≥ 3. In every dimension above 2, one can find analogues of the cube, tetrahedron and octahedron. In three dimensions, one finds two more regular polyhedra — the
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
(12-hedron) and the
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
(20-hedron) — making five
Platonic solids
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
. In four dimensions, a total of six
regular polytopes
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
exist, including the
120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...
, the
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from ...
and the
24-cell. There are no other regular polytopes, as the only regular polytopes in higher dimensions are of the
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
,
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
,
orthoplex
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
series. In all dimensions combined, there are therefore three series and five exceptional polytopes.
Moreover, the pattern is similar if non-convex polytopes are included: in two dimensions, there is a
regular star polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
for every
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
. In three dimensions, there are four
Kepler–Poinsot polyhedra, and in four dimensions, ten
Schläfli–Hess polychora; in higher dimensions, there are no non-convex regular figures.
These can be generalized to
tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
s of other spaces, especially
uniform tessellations, notably tilings of Euclidean space (
honeycombs), which have exceptional objects, and tilings of hyperbolic space. There are various exceptional objects in dimension below 6, but in dimension 6 and above, the only regular polyhedra/tilings/hyperbolic tilings are the simplex, hypercube, cross-polytope, and hypercube lattice.
Schwarz triangles
Related to tilings and the regular polyhedra, there are exceptional
Schwarz triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in .
These can be defined mor ...
s (triangles that tile the sphere, or more generally Euclidean plane or hyperbolic plane via their
triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangl ...
of reflections in their edges), particularly the
Möbius triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in .
These can be defined mor ...
s. In the sphere, there are 3 Möbius triangles (and 1 1-parameter family), corresponding to the 3 exceptional Platonic solid groups, while in the Euclidean plane, there are 3 Möbius triangles, corresponding to the 3 special triangles: 60-60-60 (
equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
),
45-45-90 (isosceles right), and
30-60-90. There are additional exceptional Schwarz triangles in the sphere and Euclidean plane. By contrast, in the hyperbolic plane, there is a 3-parameter family of Möbius triangles, and none exceptional.
Finite simple groups
The finite simple groups have been
classified into a number of series as well as 26
sporadic groups
In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.
A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. Th ...
. Of these, 20 are subgroups or subquotients of the
monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
246320597611213317192329314147 ...
, referred to as the "Happy Family", while 6 are not, and are referred to as "
pariahs
Pariah may refer to:
* A member of the Paraiyar caste in the Indian state of Tamil Nadu
* Pariah state, a country whose behavior does not conform to norms
* Outcast (person)
Science and mathematics
* Pariah dog, a type of semi-feral dog
* ''Pa ...
".
Several of the sporadic groups are related to the
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
, most notably the Conway group Co
1, which is the automorphism group of the Leech lattice, quotiented out by its center.
Division algebras
There are only three finite-dimensional associative
division algebras
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
over the reals — the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, the
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
and the
quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
. The only non-associative division algebra is the algebra of
octonions
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...
. The octonions are connected to a wide variety of exceptional objects. For example, the exceptional formally real
Jordan algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:
# xy = yx (commutative law)
# (xy)(xx) = x(y(xx)) ().
The product of two elements ''x'' and ''y'' in a Jordan alg ...
is the
Albert algebra of 3 by 3 self-adjoint matrices over the octonions.
Simple Lie groups
The
simple Lie group
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symme ...
s form a number of series (
classical Lie group
In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or ske ...
s) labelled A, B, C and D. In addition, there are the exceptional groups
G2 (the automorphism group of the octonions),
F4,
E6,
E7,
E8. These last four groups can be viewed as the symmetry groups of projective planes over O, C⊗O, H⊗O and O⊗O, respectively, where O is the octonions and the tensor products are over the reals.
The classification of Lie groups corresponds to the classification of
root systems
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often b ...
, and thus the exceptional Lie groups correspond to exceptional root systems and exceptional
Dynkin diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
s.
Supersymmetric algebras
There are a few exceptional objects with
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
. The classification of
superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
T ...
s by
Kac and Tierry-Mieg indicates that the
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ...
s G(3) in 31 dimensions and F(4) in 40 dimensions, and the Jordan superalgebras K
3 and K
10, are examples of exceptional objects.
Unimodular lattices
Up to isometry, there is only one even
unimodular lattice
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamen ...
in 15 dimensions or less — the
E8 lattice. Up to dimension
24, there is only one even unimodular lattice without
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
, the
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
. Three of the sporadic simple groups were discovered by Conway while investigating the automorphism group of the Leech lattice. For example,
Co1 is the automorphism group itself modulo ±1. The groups
Co2 and
Co3, as well as a number of other sporadic groups, arise as stabilisers of various subsets of the Leech lattice.
Codes
Some
codes
In communications and information processing, code is a system of rules to convert information—such as a letter, word, sound, image, or gesture—into another form, sometimes shortened or secret, for communication through a communication ...
also stand out as exceptional objects, in particular the perfect binary Golay code, which is closely related to the Leech lattice. The
Mathieu group
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
, one of the sporadic simple groups, is the group of automorphisms of the
extended binary Golay code, and four more of the sporadic simple groups arise as various types of stabilizer subgroup of
.
Block designs
An exceptional
block design is the
Steiner system
250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line.
In combinatorial mathematics, a Steiner system (named after Jakob Steiner) ...
S(5,8,24) whose automorphism group is the sporadic simple
Mathieu group
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...
.
The codewords of the
extended binary Golay code have a length of 24 bits and have weights 0, 8, 12, 16, or 24. This code can correct up to three errors. So every 24-bit word with weight 5 can be corrected to a codeword with weight 8. The bits of a 24-bit word can be thought of as specifying the possible subsets of a 24 element set. So the extended binary Golay code gives a unique 8 element subset for each 5 element subset. In fact, it defines S(5,8,24).
Outer automorphisms
Certain families of groups often have a certain
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
, but in particular cases, they have other exceptional outer automorphisms.
Among families of finite simple groups, the only example is in the
automorphisms of the symmetric and alternating groups In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the excepti ...
: for
the
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic prop ...
has one outer automorphism (corresponding to conjugation by an odd element of
) and the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
has no outer automorphisms. However, for
there is an
exceptional outer automorphism of
(of order 2), and correspondingly, the outer automorphism group of
is not
(the group of order 2), but rather
, the
Klein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...
.
If one instead considers
as the (isomorphic)
projective special linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associat ...
, then the outer automorphism is not exceptional; thus the exceptional-ness can be seen as due to the
exceptional isomorphism In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ''a'i'' and ''b'j'' of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such is ...
This exceptional outer automorphism is realized inside of the Mathieu group
and similarly,
acts on a set of 12 elements in 2 different ways.
Among
Lie groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
, the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
has an exceptionally large outer automorphism group (namely
), which corresponds to the exceptional symmetries of the
Dynkin diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
. This phenomenon is referred to as ''
triality
In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin( ...
.''
The exceptional symmetry of the
diagram also gives rise to the
Steinberg groups.
Algebraic topology
The
Kervaire invariant In mathematics, the Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphe ...
is an invariant of a (4''k'' + 2)-dimensional manifold that measures whether the manifold could be
surgically
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pa ...
converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. More specifically, the Kervaire invariant applies to a
framed manifold, that is, to a manifold equipped with an
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is gi ...
into
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
and a trivialization of the
normal bundle
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).
Definition
Riemannian manifold
Let (M,g) be a Riemannian m ...
. The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero. For differentiable manifolds, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. The final case of dimension 126 remains open. These five or six framed
cobordism classes of manifolds having Kervaire invariant 1 are exceptional objects related to
exotic spheres. The first three cases are related to the complex numbers, quaternions and octonions respectively: a manifold of Kervaire invariant 1 can be constructed as the product of two spheres, with its exotic framing determined by the normed division algebra.
Due to similarities of dimensions, it is conjectured that the remaining cases (dimensions 30, 62 and 126) are related to the
Rosenfeld projective plane
In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea indep ...
s, which are defined over algebras constructed from the octonions. Specifically, it has been conjectured that there is a construction that takes these projective planes and produces a manifold with nonzero Kervaire invariant in two dimensions lower, but this remains unconfirmed.
Symmetric quantum measurements
In
quantum information theory
Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information re ...
, there exist structures known as
SIC-POVM
A symmetric, informationally complete, positive operator-valued measure (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain d ...
s or SICs, which correspond to maximal sets of complex
equiangular lines In geometry, a set of lines is called equiangular if all the lines intersect at a single point, and every pair of lines makes the same angle.
Equiangular lines in Euclidean space
Computing the maximum number of equiangular lines in ''n''-dimensi ...
. Some of the known SICs—those in vector spaces of 2 and 3 dimensions, as well as certain solutions in 8 dimensions—are considered exceptional objects and called "sporadic SICs". They differ from the other known SICs in ways that involve their symmetry groups, the
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
of the numerical values of their vector components, and so forth. The sporadic SICs in dimension 8 are related to the integral octonions.
Connections
Numerous connections have been observed between some, though not all, of these exceptional objects. Most common are objects related to
8 and
24 dimensions, noting that 24 = 8 · 3. By contrast, the
pariah group
In group theory, the term pariah was introduced by Robert Griess in to refer to the six sporadic simple groups which are not subquotients of the monster group
In the area of abstract algebra known as group theory, the monster group M (als ...
s stand apart, as the name suggests.
8 and 24 dimensions
Exceptional objects related to the number 8 include the following.
* The octonions are 8-dimensional.
* The
E8 lattice can be realized as the integral octonions (up to a scale factor).
* The exceptional Lie groups can be seen as symmetries of the octonions and structures derived from the octonions; further, the E
8 algebra is related to the E
8 lattice, as the notation implies (the lattice is generated by the root system of the algebra).
* Triality occurs for Spin(8), which also connects to 8 · 3 = 24.
Likewise, exceptional objects related to the number 24 include the following.
* The Leech lattice is 24-dimensional.
* Most sporadic simple groups can be related to the Leech lattice, or more broadly the Monster.
* The exceptional
Jordan algebra
In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:
# xy = yx (commutative law)
# (xy)(xx) = x(y(xx)) ().
The product of two elements ''x'' and ''y'' in a Jordan alg ...
has a representation in terms of 24×24 real matrices together with the Jordan product rule.
These objects are connected to various other phenomena in math which may be considered surprising but not themselves "exceptional". For example, in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, 8-fold real
Bott periodicity
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable comple ...
can be seen as coming from the octonions. In the theory of
modular forms
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
, the 24-dimensional nature of the Leech lattice underlies the presence of 24 in the formulas for the
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
and the
modular discriminant
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the ...
, which connection is deepened by
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
, a development that related modular functions to the Monster group.
Physics
In
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
and superstring theory we often find that particular dimensions are singled out as a result of exceptional algebraic phenomena. For example,
bosonic string theory
Bosonic string theory is the original version of string theory, developed in the late 1960s and named after Satyendra Nath Bose. It is so called because it contains only bosons in the spectrum.
In the 1980s, supersymmetry was discovered in the co ...
requires a spacetime of dimension 26 which is directly related to the presence of 24 in the
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string ...
. Similarly, the possible dimensions of
supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
are related to the dimensions of the
division algebras
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
.
Monstrous moonshine
Many of the exceptional objects in mathematics and physics have been found to be connected to each other. Developments such as the
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
conjectures show how, for example, the
Monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order
246320597611213317192329314147 ...
is connected to
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
. The theory of
modular forms
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
shows how the algebra E
8 is connected to the Monster group. (In fact, well before the proof of the Monstrous moonshine conjecture, the
elliptic ''j''-function was discovered to encode the representations of E
8.
) Other interesting connections include how the
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
is connected via the
Golay code to the adjacency matrix of the
dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
(another exceptional object). Below is a
mind map
A mind map is a diagram used to visually organize information into a hierarchy, showing relationships among pieces of the whole. It is often created around a single concept, drawn as an image in the center of a blank page, to which associated r ...
showing how some of the exceptional objects in mathematics and mathematical physics are related.
The connections can partly be explained by thinking of the algebras as a tower of lattice
vertex operator algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usef ...
s. It just so happens that the vertex algebras at the bottom are so simple that they are isomorphic to familiar non-vertex algebras. Thus the connections can be seen simply as the consequence of some lattices being sub-lattices of others.
Supersymmetries
The
Jordan superalgebras are a parallel set of exceptional objects with
supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
. These are the
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ...
s which are related to Lorentzian lattices. This subject is less explored, and the connections between the objects are less well established. There are new conjectures parallel to the
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. ...
conjectures for these super-objects, involving different sporadic groups.
Unexceptional objects
Pathologies
"Exceptional" object is reserved for objects that are unusual, meaning rare, the exception, not for ''unexpected'' or ''non-standard'' objects. These unexpected-but-typical (or common) phenomena are generally referred to as
pathological
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in th ...
, such as
nowhere differentiable function
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
The Weierstras ...
s, or "exotic", as in
exotic spheres — there are exotic spheres in arbitrarily high dimension (not only a finite set of exceptions), and in many dimensions most (differential structures on) spheres are exotic.
Extremal objects
Exceptional objects must be distinguished from ''extremal'' objects: those that fall in a family and are the most extreme example by some measure are of interest, but not unusual in the way exceptional objects are. For example, the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
''φ'' has the simplest
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
approximation, and accordingly is most difficult to
approximate by rationals; however, it is but one of infinitely many such quadratic numbers (continued fractions).
Similarly, the (2,3,7)
Schwarz triangle
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in .
These can be defined mor ...
is the smallest hyperbolic Schwarz triangle, and the associated
(2,3,7) triangle group In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, ...
is of particular interest, being the universal
Hurwitz group
In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus ''g'' > 1, stating that the number of such automorphisms ...
, and thus being associated with the
Hurwitz curves, the maximally symmetric algebraic curves. However, it falls in a family of such triangles ((2,4,7), (2,3,8), (3,3,7), etc.), and while the smallest, is not exceptional or unlike the others.
See also
*
Exceptional isomorphism In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ''a'i'' and ''b'j'' of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such is ...
*
Exception that proves the rule
"The exception that proves the rule" is a saying whose meaning is contested. Henry Watson Fowler's ''Modern English Usage'' identifies five ways in which the phrase has been used, and each use makes some sort of reference to the role that a partic ...
References
{{reflist
Mathematical terminology